Trigonometry Laws of Sines and Cosines 5

Question 1

Heron's formula uses all three sides of a triangle. What is the semi-perimeter s of a triangle with sides 5, 12, and 13?

Accepted Answer

s = 15

Answer

The semi-perimeter is s = (5 + 12 + 13) / 2 = 30 / 2 = 15.

Question 2

In triangle ABC, the Law of Cosines gives cos A = −0.5. What is angle A?

Accepted Answer

120°

Answer

cos⁻¹(−0.5) = 120°, since cosine is negative in the second quadrant.

Question 3

Triangle MNO has M = 25°, N = 110°, and n = 30. What is the length of side m?

Accepted Answer

m = 30·sin 25° / sin 110°

Answer

By the Law of Sines, m / sin M = n / sin N, so m = n·sin M / sin N = 30·sin 25° / sin 110°.

Question 4

Which statement about the Law of Cosines is TRUE when angle C = 90°?

Accepted Answer

It reduces to the Pythagorean theorem: c² = a² + b²

Answer

When C = 90°, cos 90° = 0, so c² = a² + b² − 2ab(0) = a² + b², the Pythagorean theorem.

Question 5

An engineer measures a triangular land parcel: two sides are 80 m and 60 m, and the angle between them is 90°. What is the length of the third side?

Accepted Answer

100 m

Answer

c² = 80² + 60² − 2(80)(60)cos 90° = 6400 + 3600 − 0 = 10000, so c = 100 m.

Trigonometry Practice Test